ROmondi/STST · Datasets at Hugging Face

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6 Conclusion
Verification of probabilistic forecasts is an essential but complex step of all forecasting procedures. Scoring
rules may appear as the perfect tool to compare forecast performance since, when proper, they can simulta
neously assess calibration and sharpness. However, propriety, even if strict, does not ensure that a scoring
rule is relevant to the problem at hand. With that in mind, we agree with the recommendation of Scheuerer
and Hamill (2015) that "several different scores be always considered before drawing conclusions". This is
even more important in a multivariate setting where forecasts are characterized by more complex objects.
Weproposed a framework to construct proper scoring rules in a multivariate setting using aggregation and
transformation principles. Aggregation-and-transformation-based scoring rules can improve the conclusions
27
drawn since they enable the verification of specific aspects of the forecast (e.g., anisotropy of the dependence
structure). This has been illustrated both using examples from the literature and numerical experiments
showcasing different settings. Moreover, we showed that the aggregation and transformation principles can
be used to construct scoring rules that are proper, interpretable, and not affected by the double-penalty
effect. This could be a starting point to help bridging the gap between the proper scoring rule community
and the spatial verification tools community.
As the interest for machine learning-based weather forecast is increasing (see, e.g., Ben Bouallègue et al.
2024a), multiple approaches have performance comparable to ECMWF deterministic high-resolution fore
casts (Keisler, 2022; Pathak et al., 2022; Bi et al., 2023; Lam et al., 2022; Chen et al., 2023). The natural
extension to probabilistic forecast is already developing and enabled by publicly available benchmark datasets
such as WeatherBench 2 (Rasp et al., 2024). Aggregation-and-transformation-based methods can help ensure
that parameter inference does not hedge certain important aspects of the multivariate probabilistic forecasts.
There seems to be a trade-off between discrimination ability and strict propriety. Discrimination ability
comes from the ability of scoring rules to differentiate misspecification of certain characteristics. By defi
nition, the expectation of strictly proper scoring rules is minimized when the probabilistic forecast is the
true distribution. Nonetheless, it does not guarantee that this global minimum is steep in any misspecifi
cation direction. However, interpretable scoring rules can discriminate the misspecification of their target
characteristic. Should scoring rules discriminating any misspecification be pursued? Or should interpretable
scoring rules discriminating a specific type of misspecification be used instead?
Acknowledgments
The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference
ANR-20-CE40-0025-01 (T-REX project) and the Energy-oriented Centre of Excellence II (EoCoE-II), Grant
Agreement 824158, funded within the Horizon2020 framework of the European Union. Part of this work was
also supported by the ExtremesLearning grant from 80 PRIME CNRS-INSU and this study has received
funding from Agence Nationale de la Recherche- France 2030 as part of the PEPR TRACCS program under
grant number ANR-22-EXTR-0005 and the ANR EXSTA.
Sam Allen is thanked for fruitful discussions during the preparation of this manuscript.
References
Paolo Agnolucci, Chrysanthi Rapti, Peter Alexander, Vincenzo De Lipsis, Robert A. Holland, Felix Eigen
brod, and Paul Ekins. Impacts of rising temperatures and farm management practices on global yields
of 18 crops. Nature Food, 1(9):562–571, September 2020. ISSN 2662-1355. https://doi.org/10.1038/
s43016-020-00148-x.
Zeina Al Masry, Romain Pic, Clément Dombry, and Chrisine Devalland. A new methodology to predict the
oncotype scores based on clinico-pathological data with similar tumor profiles. Breast Cancer Research
and Treatment, 2023. ISSN 1573-7217. https://doi.org/10.1007/s10549-023-07141-5.
Carol Alexander, Michael Coulon, Y. Han, and Xiaochun Meng. Evaluating the discrimination ability
of proper multi-variate scoring rules. Annals of Operations Research, March 2022. ISSN 1572-9338.
ERROR: type should be string, got " https://doi.org/10.1007/s10479-022-04611-9."
Sam Allen, Jonas Bhend, Olivia Martius, and Johanna Ziegel. Weighted verification tools to evaluate uni
variate and multivariate probabilistic forecasts for high-impact weather events. Weather and Forecasting,
38(3):499–516, March 2023a. ISSN 1520-0434. https://doi.org/10.1175/waf-d-22-0161.1.
Sam Allen, David Ginsbourger, and Johanna Ziegel. Evaluating forecasts for high-impact events using
transformed kernel scores. SIAM/ASA Journal on Uncertainty Quantification, 11(3):906–940, August
2023b. ISSN 2166-2525. https://doi.org/10.1137/22m1532184.
28
Sam Allen, Johanna Ziegel, and David Ginsbourger. Assessing the calibration of multivariate probabilistic
forecasts. Quarterly Journal of the Royal Meteorological Society, 150(760):1315–1335, February 2024. ISSN
1477-870X. https://doi.org/10.1002/qj.4647.
Jeffrey L. Anderson. A method for producing and evaluating probabilistic forecasts from ensemble model
integrations. Journal of Climate, 9(7):1518–1530, July 1996. ISSN 1520-0442. https://doi.org/10.
1175/1520-0442(1996)009<1518:amfpae>2.0.co;2.
Andreas Basse-O’Connor, Vytaut˙ e Pilipauskait˙ e, and Mark Podolskij. Power variations for fractional type
infinitely divisible random fields. Electronic Journal of Probability, 26(none):1– 35, 2021. https://doi.
org/10.1214/21-EJP617. URL https://doi.org/10.1214/21-EJP617.
Zied Ben Bouallègue, Mariana C. A. Clare, Linus Magnusson, Estibaliz Gascón, Michael Maier-Gerber,
Martin Janoušek, Mark Rodwell, Florian Pinault, Jesper S. Dramsch, Simon T. K. Lang, Baudouin
Raoult, Florence Rabier, Matthieu Chevallier, Irina Sandu, Peter Dueben, Matthew Chantry, and Florian
Pappenberger. The rise of data-driven weather forecasting: A first statistical assessment of machine
learning–based weather forecasts in an operational-like context. Bulletin of the American Meteorological
Society, 105(6):E864–E883, June 2024a. ISSN 1520-0477. https://doi.org/10.1175/bams-d-23-0162.
1.
Zied Ben Bouallègue, Jonathan A. Weyn, Mariana C. A. Clare, Jesper Dramsch, Peter Dueben, and Matthew
Chantry. Improving medium-range ensemble weather forecasts with hierarchical ensemble transformers.
Artificial Intelligence for the Earth Systems, 3(1), January 2024b. ISSN 2769-7525. https://doi.org/
10.1175/aies-d-23-0027.1.
Albert Benassi, Serge Cohen, and Jacques Istas. On roughness indices for fractional fields. Bernoulli, 10
(2):357– 373, 2004. https://doi.org/10.3150/bj/1082380223. URL https://doi.org/10.3150/bj/
1082380223.
Alain Berlinet and Christine Thomas-Agnan. Reproducing kernel Hilbert spaces in probability and statis
tics. Kluwer Academic Publishers, Boston, MA, 2004. ISBN 1-4020-7679-7. https://doi.org/10.1007/
978-1-4419-9096-9. URL https://doi.org/10.1007/978-1-4419-9096-9. With a preface by Persi
Diaconis.
Kaifeng Bi, Lingxi Xie, Hengheng Zhang, Xin Chen, Xiaotao Gu, and Qi Tian. Accurate medium-range
global weather forecasting with 3d neural networks. Nature, 619(7970):533–538, July 2023. ISSN 1476
4687. https://doi.org/10.1038/s41586-023-06185-3.
Mathias Blicher Bjerregård, Jan Kloppenborg Møller, and Henrik Madsen. An introduction to multivariate
probabilistic forecast evaluation. Energy and AI, 4:100058, June 2021. ISSN 2666-5468. https://doi.
org/10.1016/j.egyai.2021.100058.
David Bolin and Jonas Wallin. Local scale invariance and robustness of proper scoring rules. Statistical
Science, 38(1), feb 2023. https://doi.org/10.1214/22-sts864.
Nikos I. Bosse, Sam Abbott, Anne Cori, Edwin van Leeuwen, Johannes Bracher, and Sebastian Funk. Scoring
epidemiological forecasts on transformed scales. PLOS Computational Biology, 19(8):e1011393, August
2023. ISSN 1553-7358. https://doi.org/10.1371/journal.pcbi.1011393.
Jonas Brehmer. Elicitability and its application in risk management. July 2017. https://doi.org/10.
48550/ARXIV.1707.09604.
Jonas R. Brehmer and Kirstin Strokorb. Why scoring functions cannot assess tail properties. Electronic
Journal of Statistics, 13(2), January 2019. ISSN 1935-7524. https://doi.org/10.1214/19-ejs1622.
John Bjørnar Bremnes. Ensemble postprocessing using quantile function regression based on neural networks
and bernstein polynomials. Monthly Weather Review, 148(1):403–414, December 2019. ISSN 1520-0493.
ERROR: type should be string, got " https://doi.org/10.1175/mwr-d-19-0227.1."

6 Conclusion

Verification of probabilistic forecasts is an essential but complex step of all forecasting procedures. Scoring

rules may appear as the perfect tool to compare forecast performance since, when proper, they can simulta

neously assess calibration and sharpness. However, propriety, even if strict, does not ensure that a scoring

rule is relevant to the problem at hand. With that in mind, we agree with the recommendation of Scheuerer

and Hamill (2015) that "several different scores be always considered before drawing conclusions". This is

even more important in a multivariate setting where forecasts are characterized by more complex objects.

Weproposed a framework to construct proper scoring rules in a multivariate setting using aggregation and

transformation principles. Aggregation-and-transformation-based scoring rules can improve the conclusions

27

drawn since they enable the verification of specific aspects of the forecast (e.g., anisotropy of the dependence

structure). This has been illustrated both using examples from the literature and numerical experiments

showcasing different settings. Moreover, we showed that the aggregation and transformation principles can

be used to construct scoring rules that are proper, interpretable, and not affected by the double-penalty

effect. This could be a starting point to help bridging the gap between the proper scoring rule community

and the spatial verification tools community.

As the interest for machine learning-based weather forecast is increasing (see, e.g., Ben Bouallègue et al.

2024a), multiple approaches have performance comparable to ECMWF deterministic high-resolution fore

casts (Keisler, 2022; Pathak et al., 2022; Bi et al., 2023; Lam et al., 2022; Chen et al., 2023). The natural

extension to probabilistic forecast is already developing and enabled by publicly available benchmark datasets

such as WeatherBench 2 (Rasp et al., 2024). Aggregation-and-transformation-based methods can help ensure

that parameter inference does not hedge certain important aspects of the multivariate probabilistic forecasts.

There seems to be a trade-off between discrimination ability and strict propriety. Discrimination ability

comes from the ability of scoring rules to differentiate misspecification of certain characteristics. By defi

nition, the expectation of strictly proper scoring rules is minimized when the probabilistic forecast is the

true distribution. Nonetheless, it does not guarantee that this global minimum is steep in any misspecifi

cation direction. However, interpretable scoring rules can discriminate the misspecification of their target

characteristic. Should scoring rules discriminating any misspecification be pursued? Or should interpretable

scoring rules discriminating a specific type of misspecification be used instead?

Acknowledgments

The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference

ANR-20-CE40-0025-01 (T-REX project) and the Energy-oriented Centre of Excellence II (EoCoE-II), Grant

Agreement 824158, funded within the Horizon2020 framework of the European Union. Part of this work was

also supported by the ExtremesLearning grant from 80 PRIME CNRS-INSU and this study has received

funding from Agence Nationale de la Recherche- France 2030 as part of the PEPR TRACCS program under

grant number ANR-22-EXTR-0005 and the ANR EXSTA.

Sam Allen is thanked for fruitful discussions during the preparation of this manuscript.

References

Paolo Agnolucci, Chrysanthi Rapti, Peter Alexander, Vincenzo De Lipsis, Robert A. Holland, Felix Eigen

brod, and Paul Ekins. Impacts of rising temperatures and farm management practices on global yields

of 18 crops. Nature Food, 1(9):562–571, September 2020. ISSN 2662-1355. https://doi.org/10.1038/

s43016-020-00148-x.

Zeina Al Masry, Romain Pic, Clément Dombry, and Chrisine Devalland. A new methodology to predict the

oncotype scores based on clinico-pathological data with similar tumor profiles. Breast Cancer Research

and Treatment, 2023. ISSN 1573-7217. https://doi.org/10.1007/s10549-023-07141-5.

Carol Alexander, Michael Coulon, Y. Han, and Xiaochun Meng. Evaluating the discrimination ability

of proper multi-variate scoring rules. Annals of Operations Research, March 2022. ISSN 1572-9338.

ERROR: type should be string, got " https://doi.org/10.1007/s10479-022-04611-9."

Sam Allen, Jonas Bhend, Olivia Martius, and Johanna Ziegel. Weighted verification tools to evaluate uni

variate and multivariate probabilistic forecasts for high-impact weather events. Weather and Forecasting,

38(3):499–516, March 2023a. ISSN 1520-0434. https://doi.org/10.1175/waf-d-22-0161.1.

Sam Allen, David Ginsbourger, and Johanna Ziegel. Evaluating forecasts for high-impact events using

transformed kernel scores. SIAM/ASA Journal on Uncertainty Quantification, 11(3):906–940, August

2023b. ISSN 2166-2525. https://doi.org/10.1137/22m1532184.

28

Sam Allen, Johanna Ziegel, and David Ginsbourger. Assessing the calibration of multivariate probabilistic

forecasts. Quarterly Journal of the Royal Meteorological Society, 150(760):1315–1335, February 2024. ISSN

1477-870X. https://doi.org/10.1002/qj.4647.

Jeffrey L. Anderson. A method for producing and evaluating probabilistic forecasts from ensemble model

integrations. Journal of Climate, 9(7):1518–1530, July 1996. ISSN 1520-0442. https://doi.org/10.

1175/1520-0442(1996)009<1518:amfpae>2.0.co;2.

Andreas Basse-O’Connor, Vytaut˙ e Pilipauskait˙ e, and Mark Podolskij. Power variations for fractional type

infinitely divisible random fields. Electronic Journal of Probability, 26(none):1– 35, 2021. https://doi.

org/10.1214/21-EJP617. URL https://doi.org/10.1214/21-EJP617.

Zied Ben Bouallègue, Mariana C. A. Clare, Linus Magnusson, Estibaliz Gascón, Michael Maier-Gerber,

Martin Janoušek, Mark Rodwell, Florian Pinault, Jesper S. Dramsch, Simon T. K. Lang, Baudouin

Raoult, Florence Rabier, Matthieu Chevallier, Irina Sandu, Peter Dueben, Matthew Chantry, and Florian

Pappenberger. The rise of data-driven weather forecasting: A first statistical assessment of machine

learning–based weather forecasts in an operational-like context. Bulletin of the American Meteorological

Society, 105(6):E864–E883, June 2024a. ISSN 1520-0477. https://doi.org/10.1175/bams-d-23-0162.

1.

Zied Ben Bouallègue, Jonathan A. Weyn, Mariana C. A. Clare, Jesper Dramsch, Peter Dueben, and Matthew

Chantry. Improving medium-range ensemble weather forecasts with hierarchical ensemble transformers.

Artificial Intelligence for the Earth Systems, 3(1), January 2024b. ISSN 2769-7525. https://doi.org/

10.1175/aies-d-23-0027.1.

Albert Benassi, Serge Cohen, and Jacques Istas. On roughness indices for fractional fields. Bernoulli, 10

(2):357– 373, 2004. https://doi.org/10.3150/bj/1082380223. URL https://doi.org/10.3150/bj/

1082380223.

Alain Berlinet and Christine Thomas-Agnan. Reproducing kernel Hilbert spaces in probability and statis

tics. Kluwer Academic Publishers, Boston, MA, 2004. ISBN 1-4020-7679-7. https://doi.org/10.1007/

978-1-4419-9096-9. URL https://doi.org/10.1007/978-1-4419-9096-9. With a preface by Persi

Diaconis.

Kaifeng Bi, Lingxi Xie, Hengheng Zhang, Xin Chen, Xiaotao Gu, and Qi Tian. Accurate medium-range

global weather forecasting with 3d neural networks. Nature, 619(7970):533–538, July 2023. ISSN 1476

4687. https://doi.org/10.1038/s41586-023-06185-3.

Mathias Blicher Bjerregård, Jan Kloppenborg Møller, and Henrik Madsen. An introduction to multivariate

probabilistic forecast evaluation. Energy and AI, 4:100058, June 2021. ISSN 2666-5468. https://doi.

org/10.1016/j.egyai.2021.100058.

David Bolin and Jonas Wallin. Local scale invariance and robustness of proper scoring rules. Statistical

Science, 38(1), feb 2023. https://doi.org/10.1214/22-sts864.

Nikos I. Bosse, Sam Abbott, Anne Cori, Edwin van Leeuwen, Johannes Bracher, and Sebastian Funk. Scoring

epidemiological forecasts on transformed scales. PLOS Computational Biology, 19(8):e1011393, August

2023. ISSN 1553-7358. https://doi.org/10.1371/journal.pcbi.1011393.

Jonas Brehmer. Elicitability and its application in risk management. July 2017. https://doi.org/10.

48550/ARXIV.1707.09604.

Jonas R. Brehmer and Kirstin Strokorb. Why scoring functions cannot assess tail properties. Electronic

Journal of Statistics, 13(2), January 2019. ISSN 1935-7524. https://doi.org/10.1214/19-ejs1622.

John Bjørnar Bremnes. Ensemble postprocessing using quantile function regression based on neural networks

and bernstein polynomials. Monthly Weather Review, 148(1):403–414, December 2019. ISSN 1520-0493.

ERROR: type should be string, got " https://doi.org/10.1175/mwr-d-19-0227.1."